A way to build a conformal boundary of a spacetime based on light - PowerPoint PPT Presentation

A way to build a conformal boundary of a spacetime based on light rays: the 3–dimensional case. Alfredo Bautista Santa–Cruz (UC3M-UAM) co-work with Alberto Ibort (ICMAT-UC3M) and Javier Lafuente (UCM) ICMAT – Madrid, 9/3/2018 60 years Alberto Ibort Fest Classical and Quantum Physics: Geometry, Dynamics & Control A. Bautista ( ) L –boundary 1 / 47

Contents 1 Introduction. 2 The space of light rays. 3 The L –boundary for dimension m = 3. 4 L –extensions for dimension m = 3. A. Bautista ( ) L –boundary 2 / 47

Introduction 1 Introduction. 2 The space of light rays. 3 The L –boundary for dimension m = 3. 4 L –extensions for dimension m = 3. A. Bautista ( ) L –boundary 3 / 47

Introduction R. Low proposed a real geometry based in the space of light rays N ([Low ’88, ’89, ’90, ’93, ’01, ’06]) as a generalization of the twistor geometry ([Penrose ’77, et al ’88]). A. Bautista ( ) L –boundary 4 / 47

Introduction R. Low proposed a real geometry based in the space of light rays N ([Low ’88, ’89, ’90, ’93, ’01, ’06]) as a generalization of the twistor geometry ([Penrose ’77, et al ’88]). Aim Construction and characterization of the conformal boundary suggested by R. Low, called L –boundary ([Low ’06]), for 3–dimensional spacetimes. A. Bautista ( ) L –boundary 4 / 47

Introduction Starting point ( M , C g ) conformal (Lorentz) manifold. A. Bautista ( ) L –boundary 5 / 47

Introduction Starting point ( M , C g ) conformal (Lorentz) manifold. M m –dimensional Hausdorff differentiable manifold, ( m ≥ 3). g Lorentz metric in M , ( − + + + . . . +). ( M , g ) time–oriented. � � g = e 2 σ g : σ ∈ F ( M ) C g = conformal (Lorentz) structure in M . A. Bautista ( ) L –boundary 5 / 47

Introduction Causality Given 0 � = v ∈ T p M , then v is said to be timelike ⇐ ⇒ g ( v , v ) < 0 null or lightlike ⇐ ⇒ g ( v , v ) = 0 spacelike ⇐ ⇒ g ( v , v ) > 0 causal ⇐ ⇒ timelike or null A. Bautista ( ) L –boundary 6 / 47

Introduction Causality Given 0 � = v ∈ T p M , then v is said to be timelike ⇐ ⇒ g ( v , v ) < 0 null or lightlike ⇐ ⇒ g ( v , v ) = 0 spacelike ⇐ ⇒ g ( v , v ) > 0 causal ⇐ ⇒ timelike or null Causal character can be extended to differentiable curves γ : I → M depending on γ ′ ( s ). A. Bautista ( ) L –boundary 6 / 47

Introduction Causality Given 0 � = v ∈ T p M , then v is said to be timelike ⇐ ⇒ g ( v , v ) < 0 null or lightlike ⇐ ⇒ g ( v , v ) = 0 spacelike ⇐ ⇒ g ( v , v ) > 0 causal ⇐ ⇒ timelike or null Causal character can be extended to differentiable curves γ : I → M depending on γ ′ ( s ). Causality is conformal. A. Bautista ( ) L –boundary 6 / 47

Introduction Figure: Causal character. A. Bautista ( ) L –boundary 7 / 47

The space of light rays 1 Introduction. 2 The space of light rays. 3 The L –boundary for dimension m = 3. 4 L –extensions for dimension m = 3. A. Bautista ( ) L –boundary 8 / 47

The space of light rays N Given g = e 2 σ g ∈ C g , and a differentiable curve γ : I → M , it is known that γ is null g –geodesic = ⇒ γ is null g –pregeodesic. ([Kulkarni ’88]) A. Bautista ( ) L –boundary 9 / 47

The space of light rays N Given g = e 2 σ g ∈ C g , and a differentiable curve γ : I → M , it is known that γ is null g –geodesic = ⇒ γ is null g –pregeodesic. ([Kulkarni ’88]) The space of light rays N N = { Im ( γ ) : γ is a null geodesic for some g ∈ C g } A. Bautista ( ) L –boundary 9 / 47

The space of light rays N Given g = e 2 σ g ∈ C g , and a differentiable curve γ : I → M , it is known that γ is null g –geodesic = ⇒ γ is null g –pregeodesic. ([Kulkarni ’88]) The space of light rays N N = { Im ( γ ) : γ is a null geodesic for some g ∈ C g } This is NOT true for timelike or spacelike geodesics. A. Bautista ( ) L –boundary 9 / 47

The space of light rays N Given g = e 2 σ g ∈ C g , and a differentiable curve γ : I → M , it is known that γ is null g –geodesic = ⇒ γ is null g –pregeodesic. ([Kulkarni ’88]) The space of light rays N N = { Im ( γ ) : γ is a null geodesic for some g ∈ C g } This is NOT true for timelike or spacelike geodesics. A light ray γ ∈ N can be seen as an unparametrized null geodesic. The definition of N is conformal. A. Bautista ( ) L –boundary 9 / 47

Differentiable structure of N Strong causality, ([Minguzzi, S´ anchez ’08]) M is strongly causal in x ∈ M if and only if there exists a neighbourhood basis at x of globally hyperbolic, causally convex (and convex normal) open sets. A. Bautista ( ) L –boundary 10 / 47

Differentiable structure of N Strong causality, ([Minguzzi, S´ anchez ’08]) M is strongly causal in x ∈ M if and only if there exists a neighbourhood basis at x of globally hyperbolic, causally convex (and convex normal) open sets. Figure: How can a light ray be defined by a basic neighbourhood? A. Bautista ( ) L –boundary 10 / 47

Differentiable structure of N ( M , C g ) strongly causal. A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . � � v ∈ � N = TM : g ( v , v ) = 0 . A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . � � . N = N + ∪ N − . v ∈ � N = TM : g ( v , v ) = 0 A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . � � . N = N + ∪ N − . v ∈ � N = TM : g ( v , v ) = 0 V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . � � . N = N + ∪ N − . v ∈ � N = TM : g ( v , v ) = 0 V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . N + ( C ) = { v ∈ N + : π ( v ) ∈ C } . (Null vectors at C ). A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . � � . N = N + ∪ N − . v ∈ � N = TM : g ( v , v ) = 0 V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . N + ( C ) = { v ∈ N + : π ( v ) ∈ C } . (Null vectors at C ). PN ( C ) = { [ v ] = span { v } : v ∈ N + ( C ) } . (Null directions at C ). The topology and the differentiable structure of N is inherited from PN ( C ) by the diffeomorphism γ : PN ( C ) → N V given by γ ([ v ]) = γ [ v ] A. Bautista ( ) L –boundary 11 / 47

Differentiable structure of N ( M , C g ) strongly causal. TM tangent bundle. � TM = TM − { 0 } . � � . N = N + ∪ N − . v ∈ � N = TM : g ( v , v ) = 0 V ⊂ M basic neighbourhood of x ∈ M contained in a coordinate chart with Cauchy surface C ⊂ V . N + ( C ) = { v ∈ N + : π ( v ) ∈ C } . (Null vectors at C ). PN ( C ) = { [ v ] = span { v } : v ∈ N + ( C ) } . (Null directions at C ). The topology and the differentiable structure of N is inherited from PN ( C ) by the diffeomorphism γ : PN ( C ) → N V given by γ ([ v ]) = γ [ v ] Locally, N can be seen as a bundle of spheres: N V ≃ C × S m − 2 . A. Bautista ( ) L –boundary 11 / 47

Topology in N Definition M is said to be null pseudo–convex if ∀ K compact ∃ K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′ . A. Bautista ( ) L –boundary 12 / 47

Topology in N Definition M is said to be null pseudo–convex if ∀ K compact ∃ K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′ . Theorem, ([Low ’90]) Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff. A. Bautista ( ) L –boundary 12 / 47

Topology in N Definition M is said to be null pseudo–convex if ∀ K compact ∃ K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′ . Theorem, ([Low ’90]) Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff. Figure: N is not Hausdorff. A. Bautista ( ) L –boundary 12 / 47

Topology in N Definition M is said to be null pseudo–convex if ∀ K compact ∃ K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′ . Theorem, ([Low ’90]) Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff. Figure: M is not null pseudo–convex. A. Bautista ( ) L –boundary 13 / 47

Topology in N Definition M is said to be null pseudo–convex if ∀ K compact ∃ K ′ compact such that all segments of null geodesic with endpoints at K are contained in K ′ . Theorem, ([Low ’90]) Let M be strongly causal. M null pseudo–convex ⇐ ⇒ N Hausdorff. Figure: M is not null pseudo–convex. A. Bautista ( ) L –boundary 14 / 47

Tangent space T γ N How can T γ N be described with elements of M ? A. Bautista ( ) L –boundary 15 / 47